The structure of $k$-potents and mixed Jordan-power preservers on matrix algebras

Abstract: Let $M_n(\mathbb{F})$ denote the algebra of $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic different from $2$. For $n \ge 2$, we classify all maps $φ: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ satisfying the mixed Jordan-power identity $$ φ(A^{k} \circ B) = φ(A)^{k} \circ φ(B), \quad \text{for all } A,B \in M_n(\mathbb{F}), $$ where $\circ$ denotes the (normalized) Jordan product $A \circ B := \tfrac{1}{2}(AB + BA)$ and $k \in \mathbb{N}$. We show that every such map is either constant, taking a fixed $(k+1)$-potent value, or there exist an invertible matrix $T \in M_n(\mathbb{F})$, a ring monomorphism $ω: \mathbb{F} \to \mathbb{F}$, and a $k$-th root of unity $\varepsilon \in \mathbb{F}$ such that $φ$ takes one of the forms $$ φ(X) = \varepsilon\, T\, ω(X)\, T^{-1} \quad \text{ or } \quad φ(X) = \varepsilon\, T\, ω(X)^{t}\, T^{-1}, $$ where $ω(X)$ denotes the matrix obtained by applying $ω$ entrywise to $X$, and $(\cdot)^{t}$ denotes matrix transposition. In particular, every nonconstant solution is necessarily additive. The classification relies fundamentally on the preservation of $(k+1)$-potents and their intrinsic structural properties.

• The paper establishes a full classification of maps preserving mixed Jordan products, showing nonconstant maps are additively structured and conjugate to automorphisms.
• It leverages properties of (k+1)-potents and partial orders to extend classical results on Jordan multiplicative maps.
• The results imply that any mixed Jordan-power preserver is either constant at a fixed (k+1)-potent or implemented via field automorphisms with conjugation.

Structure of $k$-Potents and Mixed Jordan-Power Preservers on Matrix Algebras

Background and Motivation

The paper investigates the structure and classification of maps on full matrix algebras over algebraically closed fields $\mathbb{F}$ with $\operatorname{char}(\mathbb{F}) \ne 2$ that preserve a family of "mixed" Jordan-type products. Specifically, it generalizes the Jordan multiplicative map theory by focusing on preservers $\phi: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ satisfying

$$\phi(A^k \circ B) = \phi(A)^k \circ \phi(B) \quad \forall A, B \in M_n(\mathbb{F}),$$

where $A \circ B = \frac12(AB + BA)$ denotes the normalized Jordan product and $k \in \mathbb{N}$.

Classical results in this area, such as those by Rickart, Johnson, Martindale, and Jodeit–Lam, have established the rigidity of multiplicative and Jordan multiplicative maps, especially regarding automatic additivity and their strong algebraic regularity. Recent work has extended these investigations to variants of the Jordan product, and the current result provides an encompassing classification for the parameterized family of "mixed Jordan–power" preservers.

Main Results and Contributions

The central result of the paper is the full classification of all maps $\phi: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ preserving the "mixed" product

$(A, B) \mapsto A^k \circ B,$

for all $n \geq 2$, $k \in \mathbb{N}$, and $\operatorname{char}(\mathbb{F}) \ne 2$. The authors show that all such maps fall into one of two classes:

1. Constant Maps: $\phi$ is constant, and its image is a fixed $(k+1)$-potent matrix.
2. (Generalized) Ring Automorphism Implementations: There exist an invertible $T \in M_n(\mathbb{F})$, a ring monomorphism $\omega:\mathbb{F} \to \mathbb{F}$, and a $k$-th root of unity $\varepsilon \in \mathbb{F}$ such that for all $X \in M_n(\mathbb{F})$,

$$\phi(X) = \varepsilon\, T\,\omega(X)\,T^{-1} \quad \text{or} \quad \phi(X) = \varepsilon\, T\,\omega(X)^t\,T^{-1},$$

where $\omega(X)$ is the matrix obtained by applying $\omega$ entrywise and ${}^t$ denotes transposition.

Notably, every nonconstant solution is additive and, in effect, a composition of a field automorphism (possibly twisted by transposition and roots of unity) and a conjugation automorphism. The proof leverages the preservation of $(k+1)$-potent elements, their associated partial orders, and orthoadditivity—a higher-order generalization of idempotent preservation and additivity in the Jordan context.

Technical Approach

The authors generalize and extend the techniques from Molnár, Lu, An–Hou, Ji, and their earlier result on Jordan multiplicative maps. Key mechanisms include:

• Reduction to $(k+1)$-potent Preservers: Any $\phi$ preserving the mixed product necessarily preserves $(k+1)$-potents, their ranks, and the orthoadditivity relation on mutually orthogonal families.
• Characterization of Partial Orders: The partial order $\preceq$ on the $(k+1)$-potents, generalizing the standard order on idempotents, is essential for propagating structural information under the preserver $\phi$.
• Automatic Additivity: By exploiting orthoadditivity for mutually orthogonal families of $(k+1)$-potents, the authors deduce that any nonconstant preserver is necessarily additive on the span of $(k+1)$-potents, then—via an iterative argument using recursive products—on the whole algebra.
• Extension to General Elements: The proof employs combinatorial and algebraic closure arguments to show that products and conjugations of special elements (including diagonals, idempotents, and powers) generate the entire algebra, allowing complete extension of the classification.
• Complete Nonlinearity for Maps into Strictly Smaller Matrix Algebras: If the codomain dimension $m<n$, the only possible preservers are constant maps taking a fixed $(k+1)$-potent value.

Strong Claims and Structural Phenomena

• Rigidity and Automatic Additivity: The classification demonstrates that any nonconstant map preserving the mixed Jordan–power structure is forced to be additive, and, up to field automorphism, conjugate to the identity or transpose. This extends the rigidity phenomena previously known for Jordan and ring multiplicative maps.
• Absence of Pathological Non-Additive Maps: The result precludes the existence of "pathological" mixed-product-preserving maps, even without assuming linearity or bijectivity.
• Diagonalizability of $(k+1)$-Potents: The note that when $\operatorname{char}(\mathbb{F})$ divides $k$, $(k+1)$-potents may be non-diagonalizable, requiring refined arguments relative to the $k=1$ (idempotent) case.

Practical and Theoretical Implications

On the practical side, this classification has consequences for preserver problems in matrix and operator algebra, functional identities, and quantum algebra, where the Jordan product arises from observables. Theoretically, it further cements the tight interconnection between multiplicative identities (even in non-linear settings), spectral properties, and the rigidity principle in algebraic structures:

• Operator Algebras and Quantum Theory: The normalized Jordan product models physical observables. The result implies that automorphism-like transformations preserving higher-order symmetrized products are extremely restricted.
• Preserver Theory: The work generalizes several classical preserver results and fully settles the mixed Jordan–power case for matrix algebras over algebraically closed fields of $\operatorname{char} \ne 2$.
• Matrix Theory: The rank and partial order properties of $(k+1)$-potents established here may be of independent interest for structural investigations in linear algebra.

Future Directions

Possible directions include:

• Extension to more general nonassociative algebras and non-algebraically closed fields, especially when $\operatorname{char}(\mathbb{F})=2$ or divides $k$.
• Variants with constraints on field automorphisms, structure-preserving maps in operator algebras, or general Jordan algebras (special and exceptional types).
• Multilinear or higher-arity product preservers.

Conclusion

The paper achieves a comprehensive classification of all mixed Jordan–power preservers on full matrix algebras over algebraically closed fields of characteristic not two. It demonstrates that the only such nonconstant maps are inner or conjugate automorphisms (potentially twisted by entrywise field automorphisms and transposition) scaled by roots of unity, or otherwise are constant to a $(k+1)$-potent. The result highlights a sharp rigidity even in the absence of linearity, extending the scope of preserver theory in matrix and operator algebra contexts.